Heavy quark κ and jet q^ transport coefficients in the Glasma early stage of heavy-ion collisions

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Heavy quark κ and jet q^ transport coefficients in the Glasma early stage of heavy-ion collisions © 2024 the Authors Published version Avramescu, Dana; Băran, Virgil; Greco, Vincenzo; Ipp, Andreas; Müller, David I.; Ruggieri, Marco Avramescu, D., Băran, V., Greco, V., Ipp, A., Müller, D. I., & Ruggieri, M. (2024). Heavy quark κ and jet q^ transport coefficients in the Glasma early stage of heavy-ion collisions. In HardProbes2023: 11th International Conference on Hard and Electromagnetic Probes of HighEnergy Nuclear Collisions (Article 056). Sissa Medialab. POS Proceedings of Science, 438. https://doi.org/10.22323/1.438.0056 2024 PoS(HardProbes2023)056 Heavy quark 𝜿and jet ˆ𝒒transport coefficients in the Glasma early stage of heavy-ion collisions Dana Avramescu,𝑎,𝑏,∗Virgil Băran,𝑐Vincenzo Greco,𝑑,𝑒 Andreas Ipp,𝑓David Müller𝑓and Marco Ruggieri𝑑,𝑒 𝑎Department of Physics, University of Jyväskylä P.O. Box 35, 40014 University of Jyväskylä, Finland 𝑏Helsinki Institute of Physics P.O. Box 64, 00014 University of Helsinki, Finland 𝑐Faculty of Physics, University of Bucharest Atomis ,tilor 405, Măgurele, Romania 𝑑Department of Physics and Astronomy, University of Catania Via S. Sofia 64, I-95123 Catania, Italy 𝑒INFN-Laboratori Nazionali del Sud Via S. Sofia 62, I-95123 Catania, Italy 𝑓Institute for Theoretical Physics, TU Wien Wiedner Hauptstraße 8, A-1040 Vienna, Austria E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] We study the impact of the Glasma fields, used to describe the very early stage of heavy-ion collisions, on the transport of hard probes, namely heavy quarks and jets. We perform numerical simulations of the strong classical fields using techniques from real-time lattice gauge theory. The resulting fields are used as background for the classical transport of ensembles of particles, described by Wong’s equations. For this purpose, we develop a numerical solver for the transport of the probes, based on colored particle-in-cell methods. We focus on the dynamics of heavy quarks and jets in the classical colored fields. To quantify the effect of the Glasma, we extract the momentum broadening of hard probes and evaluate the anisotropy transfer from the Glasma to the probes. Lastly, we evaluate the heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma, which turn out to be large and exhibit a peak, irrespective of the particle initialization. HardProbes2023 26-31 March 2023 Aschaffenburg, Germany ∗Speaker ©Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu Introduction Framework. A widely accepted approach in our field is to model relativistic heavy-ion collisions as a sequence of distinct stages. Each stage is formulated within a suitable effective theory. In this work, we are interested in the initial stage and use the Color Glass Condensate (CGC) theory [1] to describe it. The CGC lies at the high energy limit of Quantum Chromodynamics. It is based on the observation that, as opposed to a nucleus at rest which consists only of valence quarks, at high energy, the wavefunction of a high-energy nucleus quickly becomes dominated by gluons. Large occupation numbers characterize the associated gluon fields. Thus, we may treat them as classical. The classical Yang-Mills fields which arise from the collision of two such energetic nuclei are named the Glasma [2]. Literature. In the literature, there has been increasing interest in studying whether the hard probes, which are produced early in a collision, are affected by the Glasma initial stage. A pioneer work [3] first studied the heavy quark diffusion in Glasma, followed by investigating the effect of the Glasma stage on the 𝑅𝐴𝐴 and 𝑣2puzzle [4]. In another approach, the jet transport coefficient ˆ𝑞 was extracted from Wilson loops on the lattice for light-like jets [5,6]. The intriguing result was that ˆ𝑞in the Glasma is large and exhibits a peak. Lattice results for static heavy quark transport coefficient 𝜅in an over-occupied gluon plasma [7] complemented this observation, revealing that the momentum broadening increases rapidly in the early stage. Further analytical studies [8,9] supported the picture of large transport coefficients for both heavy quarks and jets in the Glasma. Lastly, recent results using effective kinetic theory, within the bottom-up thermalization scenario, [10,11] seamlessly fill the gap between the large 𝜅and ˆ𝑞in the Glasma and smaller values from the subsequent hydrodynamics stage. This study. All the aforementioned studies suggest that the Glasma has a significant impact on the early hard probes. In our study [12], we manage to surpass certain approximations done in these previous works, at the level of either Glasma or particle transport. Our refinements consist of simulating the Glasma fields on a SU(3) lattice and implementing full particle dynamics. More concisely, we employ classical lattice gauge theory to solve the Glasma fields. These fields are then used as background for the propagation of particles. Their dynamics are evolved from Wong’s equations and are numerically solved using the colored particle-in-cell method. In our particle solver, we can consider the effect of finite quark masses, formation times and initial momenta. 1. Hard probes in Glasma Glasma. The Glasma is constructed within the CGC framework. Partons from a nucleus carry different amounts of the total momentum, quantified through the Bjorken 𝑥variable. CGC is based on the high-energy separation of scales between soft, small-𝑥partons and hard large-𝑥partons. The large-𝑥partons, mostly valence quarks, act as color sources for the many small-𝑥partons, the classical gluon fields, as described by the Yang-Mills equation D𝜇𝐹𝜇𝜈 𝐴𝜇=𝐽𝜈.(1) covariant derivative field strength tensor gluon gauge field nucleus color current 2 PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu The color component 𝑎of the current of a high-energy nucleus moving along the light-cone direction 𝑥±is 𝐽𝜇,𝑎 (𝑥−, 𝑥+, 𝑥 e) ∝ 𝛿𝜇±𝛿(𝑥∓)𝜌𝑎(𝑥 e), where 𝑥 e≡ (𝑥, 𝑦). We further employ the MV model which assumes that color charges 𝜌𝑎in large nuclei are stochastic variables. Their two-point function ⟨𝜌𝑎𝜌𝑎⟩ ∝ 𝑄2 𝑠is governed by 𝑄𝑠the saturation momentum, namely the scale at which the gluon occupation number starts to saturate. Besides numerical lattice parameters, it is the only physical parameter of the Glasma. After the collision of two such thin recoilless nuclei, the Glasma fields are produced in the forward light cone. In our approach, we switch to Milne coordinates (𝜏,𝜂), where 𝜏≡√2𝑥+𝑥−is the Milne proper time and 𝜂≡ln(𝑥+/𝑥−)/2the spatial rapidity, and impose boost-invariance of the Glasma fields, namely 𝐴𝜇(𝑥)=𝐴𝜇(𝜏, 𝑥 e). This approximation enables us to numerically solve the Glasma field equations on the lattice, discretized in terms of gauge link and plaquette variables. One important feature of the Glasma fields is that their properties are primarily dictated by the saturation scale 𝑄𝑠. At late times, of the order 𝛿𝜏free−stream ∼1/𝑄𝑠, the fields enter the freestreaming regime and become more dilute. The color electromagnetic fields arrange themselves in correlation domains of typical size 𝛿𝑥 ecorr ∼1/𝑄𝑠. Moreover, because of the beam axis, the field configurations are highly anisotropic. All these features will be imprinted in the subsequent dynamics of the probes in the Glasma fields. Probes in Glasma. The dynamics of particles in Yang-Mills fields are governed by Wong’s equations d d𝝉 𝑥𝜇= 𝑝𝜇 𝑚,D d𝝉 𝑝𝜇=2𝑔Tr n𝑄𝐹𝜇𝜈 [𝐴𝜇]o𝑝𝜈 𝑚,d d𝝉 𝑄=−i𝑔[𝐴𝜇,𝑄]𝑝𝜇 𝑚 | {z } color rotation →U∈SU(3) 𝑄(𝝉)=U(𝝉,𝝉0)𝑄(𝝉0)U†(𝝉,𝝉0) .(2) coordinate momentum mass covariant derivativeproper time coupling constant color charge gauge field They describe how the coordinate 𝑥𝜇, momenta 𝑝𝜇and color charge 𝑄of classical particles change while propagating in a background gauge field 𝐴𝜇. The momenta evolution contains, in disguise, the color Lorentz force in curvilinear coordinates. The equation involving the charge may be recast in terms of a U∈SU(3)color rotation. In our particle solver, we color rotate the charge with particle Wilson lines constructed from the underlying Glasma lattice gauge links. This procedure is inspired by the colored particle-in-cell method and has the numerical advantage of conserving colored charges algebra invariants. The color charges 𝑄=𝑄𝑎𝑇𝑎are constructed with fixed quadratic 𝑞2≡𝑄𝑎𝑄𝑎and cubic 𝑞3≡𝑑𝑎𝑏𝑐𝑄𝑎𝑄𝑏𝑄𝑐Casimir invariants, where 𝑇𝑎are SU(3) generators and 𝑑𝑎𝑏𝑐 the symmetric structure constants. The ensemble of classical color components are distributed according to the one-, twoand three-point functions ⟨𝑄𝑎⟩=0,⟨𝑄𝑎𝑄𝑏⟩=1/2and ⟨𝑄𝑎𝑄𝑏𝑄𝑐⟩=𝑑𝑎𝑏𝑐/4, in the fundamental representation. Our procedure for generating random classical color charges relies on first constructing an initial quark color charge 𝑄0=𝑄𝑎 0𝑇𝑎with desired 𝑞2and 𝑞3values and then performing random color rotations 𝑄(𝜏0)=𝑈𝑄0𝑈†with 𝑈∈SU(3)chosen according to the Haar measure. This gives the initial color vector 𝑄(𝜏0)from Eq. (2). The subsequent color rotations with particle Wilson lines automatically conserve the initially fixed Casimir invariants. More details about classical color algebras and how the color charge equation is numerically solved on the lattice are provided in [12]. 3 PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu 2. Key results Quantifying the effect of Glasma. In this study, the main quantity of interest is the momentum broadening, a measure of the momentum accumulated by a particle as it passes through the Glasma. It is computed as 𝛿𝑝2 𝑖(𝜏) ≡ 𝑝2 𝑖(𝜏) − 𝑝2 𝑖(𝜏form)and then averaged over multiple particle trajectories and Glasma background events, yielding ⟨𝛿𝑝2 𝑖(𝜏)⟩. Its derivative defines instantaneous transport coefficients at each value of the proper time, namely d d𝜏⟨𝛿𝑝2 𝑖(𝜏)⟩ ≡ (𝜅𝑖(𝜏),heavy quarks, with 𝑖∈𝐿, 𝑇 ˆ𝑞𝑖(𝜏),jets, with 𝑖∈𝑧, 𝑦 .(3) The heavy quarks are initialized with 𝑝𝑇(𝜏form)and the momentum broadening is computed along the longitudinal 𝐿and transverse 𝑇directions, evaluated with respect to the beam axis, namely ⟨𝛿𝑝2 𝐿⟩ ≡ ⟨𝛿𝑝2 𝑧⟩and ⟨𝛿𝑝2 𝑇⟩ ≡ ⟨𝛿𝑝2 𝑥⟩ + ⟨𝛿𝑝2 𝑦⟩. We choose the jets to propagate along 𝑝𝑥(𝜏form), with longitudinal and transverse momentum broadenings corresponding to ⟨𝛿𝑝2 𝑧⟩and ⟨𝛿𝑝2 𝑦⟩. Since the Glasma is anisotropic, we expect the momentum broadening of particles in the Glasma to also be anisotropic. Thus, we extract the ratio of longitudinal over transverse momentum broadenings, namely ⟨𝛿𝑝2 𝐿⟩⟨𝛿𝑝2 𝑇⟩for heavy quarks and ⟨𝛿𝑝2 𝑧⟩⟨𝛿𝑝2 𝑦⟩for jets, as a measure of quark momentum anisotropy. For particles, we use a toy model initialization. They are randomly created in the transverse plane 𝑥 e(𝜏form)with 𝜂(𝜏form)=0, at a formation time proportional to the inverse of the mass 𝜏form ∝1/(2𝑚)for heavy quarks and 𝜏form =0for jets. All particles are formed with a given initial momentum, 𝑝𝑇(𝜏form)for heavy quarks and 𝑝𝑥(𝜏form)for jets, an input parameter which is then varied. Our Glasma is suitable for central collisions of large nuclei with infinite extent. The saturation momentum is chosen as 𝑄𝑠=2 GeV. Both the Glasma and particle have periodic boundary conditions. Lattice correlators. In particular quark kinematics limits, the particle and Glasma equations decouple and the momenta evolution from Eq. (2) enables one to extract the momentum broadening solely from Glasma lattice field correlators. These cases correspond to infinitely massive and highly energetic quarks. For the very massive static quarks 𝑚→ ∞, the momentum broadening is extracted from Glasma electric field correlators as [12] 𝛿𝑝2 𝑖(𝜏)static 𝑚→∞ =𝑔2 𝜏 ∫0 d𝜏′ 𝜏 ∫0 d𝜏′′ Tr {𝐸𝑖(𝜏′)𝐸𝑖(𝜏′′)}.(4) In the case of very fast lightlike jets propagating along the 𝑥-direction with 𝑝𝑥→ ∞, the momentum broadening is extracted from parallel transported electromagnetic fields according to [6] 𝛿𝑝2 𝑖(𝜏)lightlike 𝑝𝑥→∞ =𝑔2 𝜏 ∫0 d𝜏′ 𝜏 ∫0 d𝜏′′Tr{e 𝐹𝑖(𝜏′)e 𝐹𝑖(𝜏′′)}.(5) The color force components 𝐹𝑥≡𝐸𝑥,𝐹𝑦≡𝐸𝑦−𝐵𝑧,𝐹𝑧≡𝐸𝑧+𝐵𝑦are parallel transported e 𝐹𝑖≡ U† 𝑥𝐹𝑖U𝑥with Glasma Wilson lines along the 𝑥-direction. The momentum broadening and transport coefficients derived from lattice correlators provide valuable numerical checks and a baseline of comparison for dynamical quarks simulated with our solver. 4 PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu ⟨𝛿p2⟩[GeV2] 0 2 4 6 8 ⟨𝛿pL 2⟩× 2 ⟨𝛿pT 2⟩ 𝛿𝜏 [fm/c] 0 0.5 1 1.5 2 ⟨𝛿pL 2⟩/⟨𝛿pT 2⟩ 1 2 𝛿𝜏 [fm/c] 0 0.5 1 1.5 2 𝜅[GeV2/fm] 0 10 20 30 𝜅L× 2 𝜅T Qs= 2 GeV 0 0.1 0.2 0.3 0 10 20 p T (𝜏 form ) [GeV] 0 2 5 10 𝜅L× 2 𝜅T beauty quarks quark static dynamic Figure 1: Proper time evolution in terms of 𝛿𝜏 ≡𝜏−𝜏form of (top left) longitudinal (multiplied by 2) and transverse components of the momentum broadening, (bottom left) their ratio which yields the anisotropy (right) instantaneous transport coefficients 𝜅, both longitudinal (multiplied by 2) and transverse with (right inset) a zoom-in on the very-early stage. All simulations are done for beauty quarks with 𝑚beauty ≈4.2 GeV. We compare dynamical quarks, as simulated using Wong’s equations and having fixed 𝑝𝑇(𝜏form)(colored lines) with static quarks (grey dashed lines) extracted from lattice Glasma electric fields correlators as in Eq. (4). Heavy quarks. We extract both the longitudinal ⟨𝛿𝑝2 𝐿⟩and transverse ⟨𝛿𝑝2 𝑇⟩momentum broadenings of dynamical beauty quarks with initial 𝑝𝑇(𝜏form) ∈ {0,2,5,10}GeV and compare them to the static quark case from Eq. (4). The dynamical quarks exhibit deviations from the infinite quark mass scenario, as shown in all the curves from Fig. 1. The quark momentum is anisotropic throughout the evolution, see lower left of Fig. 1and the anisotropy is ordered with the inverse of the initial 𝑝𝑇. Interestingly, static quarks which have null momentum are less anisotropic than dynamical quarks with vanishing initial 𝑝𝑇but which propagate in the Glasma fields. On the other hand, faster quarks with larger initial 𝑝𝑇become less anisotropic than the static ones. Both ⟨𝛿𝑝2 𝐿⟩and ⟨𝛿𝑝2 𝑇⟩increase rapidly at extremely early times, see top left of Fig. 1, when the heavy quarks are rapidly accelerated by the coherent electric Glasma flux tubes. As the Glasma fields expand longitudinally and become dilute, this increase is slowed down. These cause the corresponding derivatives 𝜅𝐿and 𝜅𝑇to exhibit a very large initial peak, followed by a gradual decrease, as represented in the right inset of Fig. 1. The total 𝜅peak ≈19 −22 GeV2/fm depends on the initial 𝑝𝑇and is located at 𝛿𝜏peak ≈0.03 −0.05 fm/𝑐. At late times this increase either plateaus for ⟨𝛿𝑝2 𝑇⟩, yielding a null 𝜅𝑇or intriguingly decreases for ⟨𝛿𝑝2 𝐿⟩, giving a negative 𝜅𝐿. A possible explanation would be that the longitudinal dynamics are triggered by plasmon oscillations of the Glasma fields, as similarly reported in [7]. Moreover, the heavy quark 𝜅in the Glasma is fundamentally different from the diffusion coefficient obtained in a Langevin approach. There, 𝜅 increases linearly without reaching a peak, is less steep and has smaller values than in Glasma. 5 PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu ⟨𝛿p2⟩[GeV2] 0 1 2 3 4 ⟨𝛿pz 2⟩ ⟨𝛿py 2⟩ 𝜏[fm/c] 0 0.5 1 1.5 2 ⟨𝛿pz 2⟩/⟨𝛿py 2⟩ 1 2 3 𝜏[fm/c] 0 0.5 1 1.5 2 q[GeV2/fm] 0 5 10 15 20 qz qy Qs= 2 GeV 0 0.1 0.2 0.3 5 10 15 px(𝜏form)/m 1 2 5 10 qz qy jet quarks quark lightlike dynamic Figure 2: (Top left) Momentum broadening along longitudinal 𝑧and in transverse 𝑦directions, (bottom left) jet momentum anisotropy given by their ratio, (right) jet directional transport coefficients ˆ𝑞𝑧and ˆ𝑞𝑦with (right inset) a zoom-in on extremely early times, as a function of proper time. The simulations are done for fast quarks, formed at 𝜏form =0 fm/𝑐, with various initial momenta 𝑝𝑥(𝜏form)and masses 𝑚, while the results are plotted in terms of their ratio 𝑝𝑥(𝜏form)/𝑚(colored lines). These are compared to the lightlike quarks (grey dashed lines) computed from lattice Glasma electric and magnetic field correlators, see Eq. (5). Jets. We simulate dynamical jet quarks with fixed initial 𝑝𝑥(𝜏form) ∈ {1,2,5,10}, extract the longitudinal ⟨𝛿𝑝2 𝑧⟩and transverse ⟨𝛿𝑝2 𝑦⟩momentum broadenings, and then compare them to the lightlike jet results evaluated with Eq. (5). In a similar manner as for heavy quarks, dynamical jet curves differ from the limiting case of extremely fast quarks, as may be seen in Fig. 2. The anisotropy for dynamical jets is consistently larger than the lightlike jets, see (bottom left) part of Fig. (2), but both results are highly anisotropic at the typical times when the Glasma stage would end, roughly 𝛿𝜏Glasma ≈0.3 fm/𝑐. The proper time evolution of the longitudinal and transverse jet momentum broadening is parametrically compatible with the heavy quarks, see Fig. 1, since similar underlying Glasma dynamics govern the very early and late time behaviors. Consequently, the jet transport coefficients ˆ𝑞𝑧and ˆ𝑞𝑦have a sharp increase, with a peak of total ˆ𝑞peak ≈24−30 GeV2/fm, depending on mass and initial momentum but located at the same proper time 𝜏peak ≈0.04 fm/𝑐, followed by a long-lasting decay. Similarly, at late times, the transverse ˆ𝑞𝑦is null while ˆ𝑞𝑧becomes negative, being subject to the same first Glasma plasmon oscillation. Curiously, as we report in [12], these longitudinal momentum broadening oscillations continue to persist only for heavy quarks. Summary We developed a numerical solver for heavy quarks and jets in the Glasma fields. We extracted the momentum broadening and used it to compute transport coefficients, namely 𝜅and ˆ𝑞. In the Glasma, they are large, steep and exhibit a peak. Moreover, the accumulation of momentum is 6 PoS(HardProbes2023)056 Heavy quark 𝜅and jet ˆ𝑞transport coefficients in the Glasma Dana Avramescu anisotropic. All of these quantities vary with respect to the details of particle initialization, namely mass, formation time and initial momentum. Acknowledgments. D. A. acknowledges funding from the Academy of Finland, Center of Excellence in Quark Matter project 346324. V. G. acknowledges funding from UniCT under “Linea di intervento 2” (HQCDyn Grant). D. M. acknowledges funding from the Austrian Science Fund (FWF) projects P 34455 and P 34764. We are grateful to K. Boguslavski, T. Lappi and H. Mäntysaari for many insightful discussions. References [1] F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, The Color Glass Condensate,Ann. Rev. Nucl. Part. Sci. 60 (2010) 463 [1002.0333]. [2] T. Lappi and L. McLerran, Some features of the glasma,Nucl. Phys. A 772 (2006) 200 [hep-ph/0602189]. [3] M. Ruggieri and S.K. Das, Cathode tube effect: Heavy quarks probing the glasma in p -Pb collisions,Phys. Rev. D 98 (2018) 094024 [1805.09617]. [4] Y. Sun, G. Coci, S.K. Das, S. Plumari, M. 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